Download The origins of diversity in a simple model of evolution

The implications of neutral
evolution for neutral ecology
Daniel Lawson
Bioinformatics and Statistics Scotland
Macaulay Institute, Aberdeen
How is
How
is diversity
Diversity
maintained?
maintained?
Talk OutLine
• Background: The Unified
Neutral theory of Biodiversity
and Biogeography
• Implications from an
evolution model
• Application to Ecology
• Application to Pine Trees
What is ecological neutrality?
All individuals are equivalent.
(with respect to their chances of having
offspring in the next generation)
Squawk!
Probability of occurrence: “When Pigs Fly”
Why consider neutrality at all?
• Neutrality: theory of chance events.
• High observed diversities needed
explaining, but no general theory.
• Evolution and ecology are inherently
linked.
• Need a “null model” – minimum model that
explains diversity.
The Neutral Model
• Assume all individuals are 'equal'
• Valid for Phenotypes without function
• Genotype regions not coding for protein
synthesis (12% of Human DNA is variable!
Redon et al. Nature. doi:10.1038/nature05329 )
– Each individual has the same probability to
die (pk), or give birth (pb), in a time step
• For simplicity, assume the total population (N)
has reached equilibrium (pk = pb)
• Mutations (and/or colonisation) can
occur, reproduction is (a)sexual
The neutral model
• Consider N individuals each labeled by
Birth or Colonise
species:
TIMESTEP:
•
Pick an individual (from
N) and mark it to die.
• Pick an individual (from
N) and copy it, or with
probability pm, colonise
with a new species.
• Kill the marked
individual.
Same as a mutation…
Death
Making the model work
• Common species are common at many
sites - communities don’t exist in isolation
• Exchange individuals with a
metacommunity of size Jm >> N
• Metacommunity composition changes
much more slowly that local community
Full ecological model
Local Community 1
Metacommunity
Smaller populations
Large population
Species composition
determined by migration
Evolved by mutation
Composition changes in
time
Static composition
…
Local Community n
Local Community 2
Assumptions?
•
•
•
•
Fixed population size
All individuals are equivalent
Individual life history is irrelevant
There is a speciation “event”
60
40
Jm = 100
20
Species
Abundance
species.table(a)
80
100
Results of the model
0
Pm = 0.01
0
Simulation performed using
package UNTB in R
2000
4000
6000
Time
8000
10000
Initial results
• Explains Species Abundance Distributions
• But Species Lifetimes for abundant
species in metacommunity is impossibly
long!
(longer than the history of earth for a common
species to be replaced worldwide)
‘Fixed’ ecological model
• Consider N individuals each labeled by species:
TIMESTEP:
•
Pick an individual (from
N) and mark it to die.
• Pick an individual (from
N) and copy. Prob. pa a
proportion speciate
allopatrically.
• Kill the marked
individual.
Death
FISSION SPECIATION
Mutate proportion of population allopatrically
Spatial Version
Random Death
Local
Reproduction
Local Community
• No need for metacommunity – space takes care of it!
How is diversityRelation
Species-Area
maintained?
Number of species at a
given scale.
A=1
A=4
A=9
D=2
D=6
D=8
Species Area Relation
Country or Continental
scale
Regional scale
Intercontinental scale
Power law for intermediate scales
Diversity Time Series
Normalised Diversity
100 150 200 250
300
(10000 individuals)
50
Diversity is well defined,
even though common
types change constantly
0
Type Richness
Simpson Index
0
50000
100000
Time
150000
200000
Results of extensions
• Explains the power law species area
relation – and deviations from it
• Space is a satisfying explanation of
metacommunity
• Although specific species change
constantly, diversity is well defined
• Fission solves species lifetimes problem
(but what is it?)
Success and failure
See: J Chave, Ecol. Lett. 2004
• Not good for birds (they move too much)
• Fits “non-persistent” fish species – but not
dominant species
• Good fits to rainforests in Camaroon,
Ecuador, Panama, Peru – poor on Barro
Colorado Island
• Hard to distinguish from distributions of
very specialised species in patchy terrain.
Success and failure
See: J Chave, Ecol. Lett. 2004
• Equivalence of individuals questionable
(only 26% of species in one Rainforest).
• But per-capita averages of species often
show equivalence.
IN SHORT:
It works more often than you’d expect, but
not always
Part 2: Observations.
• We expect a “species”:
• To be “different enough” from other species.
• To be constant in time. An individual of a species
today is comparable with an individual of that
species in the past.
• But how different is “different enough”?
• How constant is constant?
• These concepts aren’t in the model!
The Lineage
Type
Space
Extinct Lineages
1 ‘family’
2 ‘species’
4 ‘types’
Time
7 ‘lineages’
?
Diversity measures
• Measured diversity depends on diversity
measure:
over
• Species Richness:
DRaw = ∑1 Sum
species i
The “Number” of different types
• Simpson Diversity:
Diversity measure
accounting for different
rarities
i
1
DS =
2
∑ pi
i
Proportion of species i from total population N
• Rao Index:
Diversity measure
accounting for difference
between types
DRau = ∑ d ij pi p j
i, j
“Difference” between types
Normalised Simpson Diversity (D=28.5)
Number of Types (D=117)
(10000 individuals)
0.0
Normalised Diversity
0.5
1.0
1.5
2.0
2.5
Diversity Time Series
0
50000
100000
Time
150000
200000
Normalised Simpson Diversity
N Simpson Types
Normalised Rao Index
(10000 individuals)
All species are not “equal” with respect to
Diversity!
0.0
Normalised Diversity
0.5
1.0
1.5
2.0
2.5
Diversity Time Series
0
50000
100000
Time
150000
200000
Assumptions?
•
•
•
•
Fixed population size
All individuals are equivalent
Individual life history is irrelevant
There is a speciation “event”
Part 3:
Relation to Ecology
Phenotype Distribution
• Consider 1 dimensional case: mutations can be
either to the left or to the right.
• Expected pattern is a Normal Distribution:
Mean
Proportion of
individuals
But…
Variance
Pattern is
produced by
Selection.
Observable (Height, weight, etc)
Test Problem
Normal
Distribution?
QUESTIONS:
Are spots functional in
Imaginarius Forma?
How many types/species do
we have?
What action should be
taken to save spotty variety?
Rare Spotty Variety
An evolution model
• Consider N individuals each labeled by
phenotype position:
TIMESTEP:
• Pick an individual (from
N) and mark it to die.
• Pick an individual (from
N) and copy it. With
probability pm Mutate to
a similar type.
• Kill the marked
individual.
Test Problem
ANSWERS: Can be neutral.
- Spots might not be
functional.
- Only one species in any
real sense.
- Saving Spotty variety
requires spatial segregation
from Less Spotty variety.
Solution
• Simplify the model – consider only first two
moments of the distribution.
• Peak is a Gaussian distribution of area 1
with dynamic mean µ and width w.
● Select death location x ● Select birth location y, mutated by 1 with probability Pm
● Remove 1/N from death location and place at birth location
● Update µ and w
Solution method
• Write down equations for the change in the
mean and the variance of the peak position
µ and the width w.
• Take continuous limit to obtain Stochastic
Differential Equations.
• Solve…
Neutral Phenotype Results
• Width of peak is proportional to fluctuations in
the width of the peak.
• Corresponds to multiple clusters
• Peak position drifts with constant speed when
population size changes.
• Evolution speed is the same in small and large populations!
• Obtain an analytic solution to act as a null
hypothesis.
• Clearly, differences between types matter!
Fission Speciation
• Consider N individuals each labeled by
species:
Fission ASSUMES a
neutral drift process.
But differences can’t
change without a
death!
?
Death
FISSION SPECIATION
Mutate proportion of population allopatrically
Fix fission speciation?
Barriers
(move in
time)
Random Death
Local
Reproduction
Implications?
• There is no “natural” species definition –
though arbitrary cut-offs still work.
• (Following holds in sexual case, where there is a natural
species definition)
• No “speciation event” – but a “speciation
process”.
• Fission speciation makes little sense in
this context – and the fix is complex.
• So: neutral ecological model is not
“parsimonious” for the metacommunity.
Full ecological model
Local Community 1
Metacommunity
Smaller populations
Not well modelled by
neutral evolution!
Species composition
determined by migration
BAD NULL MODEL
Composition changes in
time
Local Community 2
Local Community n
… Local community processes
are consistent with the neutral
model
GOOD NULL MODEL
Part 4: Application to Pine Trees
• Pine trees produce varying monoterpenes.
• Large diversity observed within a forest.
• Observed forests are remnants of much larger
historical forests • Metacommunity concept relevant
• Neutrality is a good null model within a single
species.
• But monoterpenes can effect sapling mortality…
which effect is most important?
(not) A neutral model
Work with Colin Beale and Jack Lennon
• Trees grow at a given location • And compete for resources.
• Therefore future success is driven by
intensity of competition.
• Neutral model with respect to genotype,
but not individuals.
• Resolves the problem of non-observation
of individual level equivalence.
Model details
• Trees grow and compete for space
• Trees produce seeds, which disperse
• Seeds are pollinated by other trees, whilst
still on the mother tree
• Diversity of local forest is maintained by
occasional external pollen
• Monoterpene production is heritable
Colour represents
terpene
concentrations
similar colour similar terpenes recent ancestry
0.2
400
0
0.1
200
Log-Likelihood
0
-200
Max Log-likelihood
Proportion of runs more likely than data
0.00000
0.00005
0.00010
0.00015
Probability of seeds coming from outside
0.00020
Conclusions
• Neutrality is a useful concept for null
models
• Ecological models can be informed by
evolution
• Speciation “event” - examined more closely
• Null models are useful to inform which
processes are interesting
Beyond Neutrality
• Compare with other models
• Deterministic Differential Equation Models
• Stochastic models with selection
• Network models, etc.
• Neutrality is not for life – its just for
Christmas!
• Solves some problems but is just a null model!
References
Hubbell: “Unified neutral theory of biodiversity and biogeography”, 2001
Chave: “Neutral Theory and Community Ecology”,
Ecology Letters, (2004) 7: 241–253
Lawson and Jensen:
“Neutral Evolution as Diffusion in phenotype space: reproduction with mutation but without selection”
Physics Review Letters, March 07 (98, 098102)
www.arxiv.org/abs/q­bio/0609009
Package UNTB for R
Thank you for your attention!
Ecological Model Results
Number of individuals
Species number Θ = (population size)(probability of a new species) ordered by size
So what is diversity?
• Ecological Sense: “number” of different
species or types
• Requires definition of species:
• Biological Species concept?
• Phenotypically distinct?
• Genotypic species concept?
• Definition of genotypic species is arbitrary:
• Cut-off in time to “last common ancestor”
• Need a difference based measure.
Species Area Relation
2
Diversity (Log Scale)
5
10 20
50 100
Simpson Index on Species
Simpson Index on Types
Raw Species Count
Raw Type Count
Rao Index
5
10
20
50 100
Area (Log Scale)
500
2000
Test Problem
Mean
Variance
Time average is a
Normal Distribution
when measured
relative to current
mean position
- No higher order
effects, on average
Solving for the width
Mutation
distance
d (w ) =
2
Change in
variance (in a =
 * 2w
 p −
N

Generation
time

dT

2w
+
dW
N
Deterministic part
+ Noise part
2
dW is
Random,
mean 0
timestep)
Solution at steady state:
2
Npm
( Npm ) 2 w2
p( w)dw 
e dw
5
2w
Power-law decay at large w
2
Neutral Clustering results
• Mean width: w 
• Position:
x
Npm
8
 T ( pm  w ) <
2
RMS
Fluctuations in w also ~ N0.5
With time in generations...
<x>RMS is independent of N !
pmT
2
• Compare with diffusion:
x
RMS

pmT
N
w
RMS

pmT
N
• Diffusion “does nothing” in infinite populations...
evolution does “more”!
0.0000
0.0010
Variogram gamma
0.000
0.010
Variogram gamma
0.0e+00
1.0e-05
2.0e-05
Variogram gamma
tricyclene
0
0
0
20
20
20
40
Distance
a.pinene
40
Distance
b.pinene
40
Distance
60
80
60
80
60
80
[Algebra]